Based on the finite volume framework, a scale-dependent interface flux function is needed in multi-scale modeling and simulation.
Different from a purely upwind flux which loses efficiency in the collisional limit, we here develop a kinetic central-upwind flux function based on an integral solution of the kinetic model equation.
The integral solution originates from Kogan's monograph on rarefied gas dynamics [1]} and has been employed by a series of gas-kinetic schemes [2]}, [3]}, [4]}, [5]}, [6]}.
Let us rewrite the stochastic BGK equation (REF ) for the gPC coefficients vector along the characteristics,
\(\frac{D \hat{f}_\alpha }{D t} + \nu _\alpha \hat{f}_\alpha = \nu _\alpha \hat{m}_\alpha .\) 
